20. Spectral Line Shapes, Widths, and Shifts

Observed spectral lines are always broadened, partly due to the finite
resolution of the spectrometer and partly due to intrinsic physical causes. The
principal physical causes of spectral line broadening are Doppler and pressure
broadening. The theoretical foundations of line broadening are discussed in
Atomic, Molecular, & Optical Physics Handbook, Chaps. 19 and
57, ed. by G.W.F. Drake (AIP, Woodbury, NY, 1996).

Doppler Broadening

Doppler broadening is due to the thermal motion of the emitting atoms
or ions. For a Maxwellian velocity distribution, the line shape is
Gaussian; the full width at half maximum intensity (FWHM) is,
in Å,

T is the temperature of the emitters in K, and M the atomic
weight in atomic mass units (amu).

Pressure Broadening

Pressure broadening is due to collisions of the emitters with
neighboring particles [see also Atomic, Molecular, & Optical Physics
Handbook, Chaps. 19 and 57, ed. by G.W.F. Drake (AIP, Woodbury,
NY, 1996)]. Shapes are often approximately Lorentzian, i.e.,
I(λ) ∝ {1 + [(λ -
λ0)/Δλ1/2]2}-1.
In the following formulas, all FWHM's and wavelengths are expressed in Å,
particle densities N in cm-3, temperatures T in K,
and energies E or I in cm-1.

Resonance broadening (self-broadening) occurs only between identical
species and is confined to lines with the upper or lower level having an
electric dipole transition (resonance line) to the ground state. The FWHM may
be estimated as

where λ is the wavelength of the observed line.
fr and λr are the oscillator
strength and wavelength of the resonance line; gk and
gi are the statistical weights of its upper and lower levels.
Ni is the ground state number density.

Van der Waals broadening arises from the dipole interaction of an
excited atom with the induced dipole of a ground state atom. (In the case of
foreign gas broadening, both the perturber and the radiator may be in their
respective ground states.) An approximate formula for the FWHM, strictly
applicable to hydrogen and similar atomic structures only, is

where µ is the atom-perturber reduced mass in units of u, N
the perturber density, and C6 the interaction constant.
C6 may be roughly estimated as follows:
C6 = Ck - Ci,
with Ci(k) = (9.8 × 1010)
(αdR2i(k) αd in cm3,
R2 in a02). Mean atomic
polarizability αd ≈ (6.7 × 10-25)
(3IH/4E*;)2 cm3, where
IH is the ionization energy of hydrogen and E*; the
energy of the first excited level of the perturber atom.
R2i(k) ≈ 2.5
[IH/(I-Ei(k))]2, where
I is the ionization energy of the radiator. Van der Waals broadened
lines are red shifted by about one-third the size of the FWHM.

where Ne is the electron density. The half-width parameter
α1/2 for the Hβ line at 4861 Å, widely used
for plasma diagnostics, is tabulated in the table below for some typical
temperatures and electron densities [33]. This
reference also contains α1/2
parameters for other hydrogen lines, as well as Stark width and shift data for
numerous lines of other elements, i.e., neutral atoms and singly charged ions
(in the latter, Stark widths and shifts depend linearly on Ne).
Other tabulations of complete hydrogen Stark profiles exist.

Values of Stark-broadening parameter α1/2 for the
Hβ line of hydrogen (4861 Å) for various temperatures
and electron densities.